244 ANALYTICAL MECHANICS 7. Two trains, weighing 150 tons each and moving towards each other at the rate of 40 miles an hour, collide. Find the average force which comes into play if the collision lasts 1.5 seconds. 8. A body explodes while at rest and flies to pieces. If at any instant after the explosion the different parts of the body are suddenly connected, will it move? 9. A shell of mass m explodes at the highest point of its flight and breaks into two parts, the one n times the other. Find the velocity of one piece if the other is brought to rest for an instant by the explosion. The velocity of the shell at the instant of explosion is v. 10. In the preceding problem will the motion of the center of mass of the entire shell be affected by the explosion? Answer this question on the assumption (a) that there is no air resistance, (b) that there is an air resistance. 11. A man walks from one end to the other of a plank placed on a smooth horizontal plane. Show that the plank is displaced a distance M 7 M+m where M and m are the masses of the man and of the plank, respectively, and I is the length of the plank. 12. A shell, which weighs 150 pounds, strikes an armor plate with a velocity of 2000 feet per second and emerges on the other side with a velocity of 500 feet per second. Supposing the resisting force to be uniform, find its magnitude and show that the impulse produced by it equals the change in the momentum of the shell while plowing through the plate. The plate is 10 inches thick. COLLISION AND IMPACT. 197* Central Collision. —- If two bodies collide while moving along the line which joins their centers of mass the collision is said to be central. In order to fix our ideas suppose the colliding bodies to be spheres, then Fig. 118 represents roughly the state of affairs during the collision. For a short interval of time after the spheres come into contact their centers approach each other and a little deformation takes place in the neighborhood of the point of contact at the end of which the centers of the spheres are, just for an instant, at rest with respect to one another, and are moving with a